How to find an irrational number in <mrow

How to find an irrational number in $\mathbb{Q}\cap \left[0,1\right]$?
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Kaylyn Ewing
For an "explicit" construction, consider $z=\sum _{i=1}^{\mathrm{\infty }}{2}^{-{k}_{i}}$ , where given ${k}_{1}<\dots <{k}_{n}$, if $\sum _{i=1}^{n}{2}^{-{k}_{i}}={a}_{m\left(n\right)}$. Note that ${a}_{m\left(n\right)} so z∈Vm(n). Since $m\left(n\right)\to \mathrm{\infty }$, $z$ is in all the ${V}_{i}$. Its base-2 expansion contains infinitely many 1's but also has arbitrarily large gaps between the 1's, so can't be eventually periodic, and thus $z$ must be irrational.